heat absorption by a thermal system

Heat Absorption by a Thermal System

Zdzisław Pluta, Tadeusz Hryniewicz*

Koszalin University of Technology, Raclawicka 15-17, PL 75-620 Koszalin, Poland

*E-mail address: [email protected]

„Rem tene, verba sequentur”

Katon the Elder

ABSTRACT

The work deals with revealing the quantum nature of the heat absorption phenomenon by a

thermal system, describing the phenomenon with a proper energetic interpretation. At first, the existent

description of the heat absorption phenomenon by a system has been described. Then the mechanical-

thermal analogy was introduced, using it for a better explanation of the determined thermal notions. A

set of characteristics of the heat absorption phenomenon was developed by taking advantage of this

analogy. The starting point from the source thermal characteristics, being differential dependence of

thermal force on temperature, has been regarded. That made it possible to achieve a detail solution,

being the reason model of the described natural system, which is the model having a physical sense.

Keywords: Heat; Quantum nature of the reality; Temperature; Specific heat; Heat force; Heat work;

Heat potential; Potential field; Temperature constant; Thermal energy

1. INTRODUCTION

At the very beginning it is worth notifying that there are many phenomena possessing a

quantum character. That means the magnitudes describing them change their values between

the neighbouring constant values. These constant values characterize the system states where

the determined phenomenon occurs. Thus non-continuous, quantum change of the state values

takes place, and the continuity of changes touches only the interstate values of the magnitudes

describing the studied phenomenon.

It results that one should clearly indicate the appearing layers and surfaces of the natural

system. These layers are the determined interstate spaces, and the surfaces are the potential

fields. In the spaces everything undergoes changes but on the fields the variability does not

occur. One should consider them also on the description plane of natural phenomena. That is a

pity, the superficial, simplified, linearized descriptions of the natural systems are usually

presented, thus disregarding their true, quantum nature.

One of such phenomena of the quantum nature is the phenomenon of heat absorption by

a thermal system. Unfortunately, until now it has not been described adequately. The existent

descriptions are so simplified that they made it impossible to recognize the quantum nature of

the phenomenon. This is why there is no mentioning about the existence of heat energy, and

instead the heat is defined as the energy.

International Letters of Chemistry, Physics and Astronomy Online: 2013-01-10ISSN: 2299-3843, Vol. 6, pp 17-32doi:10.18052/www.scipress.com/ILCPA.6.17CC BY 4.0. Published by SciPress Ltd, Switzerland, 2013

This paper is an open access paper published under the terms and conditions of the Creative Commons Attribution license (CC BY)(https://creativecommons.org/licenses/by/4.0)

https://doi.org/10.18052/www.scipress.com/ILCPA.6.17

That work is to explain the quantum nature of the heat absorption phenomenon by the

thermal system; it adequately describes that phenomenon, by providing with it a proper,

energetic interpretation. Heat and thermal energy are here clearly differentiated.

All that results from the source starting point, the key link of the action chain. This link

forms a general, differential record of the phenomena occurring at the same time during the

heat exchange in a thermal system.

2. EXISTENT DESCRIPTION OF THE HEAT ABSORPTION PHENOMENON BY

A SYSTEM

The existent description of this phenomenon should be begun from the literature, on up-

to-date explanation of the heat magnitude absorbed by the system. That heat has not been

defined univocally and clearly. For instance, the literature [1] provides the definition of heat

which is the energy that the system of higher temperature transfers to the system of lower

temperature, through the other one being in contact, in the process of getting/approaching of

both systems to a thermal equilibrium. According to such an interpretation of heat (identified

with the energy), that energy varies in a continuous manner. No surprising then that the

quantum form of the phenomenon has not been noticed.

Literature [2] also states that heat is the energy. The work [3] discusses the heat

exchange. It interprets this as the phenomenon appearing in case if the temperature difference

inside a system or between some systems, being in a position able to react one on each other,

may occur. However, it does not define the heat.

In the work [4], there are several different explanations of the heat notion. Moreover, in

the past there was a hypothesis that heat is a sort of weightless substance. Anton Laurent

Lavoisier (1789) called it the calorium. Julius Robert Mayer (1842) was probably to prove

that the heat is also a sort of energy. In turn, Helmholtz called the heat of such a part of

energy, which may be transferred from one body to another thanks to the temperature

difference of both bodies.

One may note, there is no agreement in the opinions regarding the content of the heat

notion. The presented standpoints are not uniform, but divergent. In practice it appears, none

of the explanations has an adequate character.

The total heat absorbed by a body is determined by the following formula:

122

1ttmcQ

t

tc (1)

where: m – body mass, 12 tt – temperature increment, that is the difference between the

final 2t and initial 1t temperature of the body.

In reference to the amount of substance, the equation (1) assumes the form

122

1ttc

m

Qq

t

tc

c (2)

where the symbol cq denotes the unit heat. That way the magnitude is called in the literature

covering thermodynamics [4], as well as that connected with heat exchange [3]. The

magnitude

18 ILCPA Volume 6

1212

12

1 tt

q

ttm

Qc cct

t

(3)

has been called the average specific heat which absorbs the delivered heat.

In accordance with the literature [6] the specific heat established for the infinitively

small the temperature increment dt has been called the real specific heat (real heat capacity):

dt

dqc c (4)

Further considerations cover the conditions determining the unit heat cq . For instance,

the references [4, 6] mention, that if the dependence tcc is given, then the heat could be

determined from the equation

2

1

t

t

c dttcq (5)

On the other hand, if the dependence of specific heat on temperature was given, for

instance in the form of a function

2

0 ttctc (6)

then

3

1

3

2

2

1

2

212032

ttttttcqc

(7)

Next, if the dependence tc is known in the form of a plot, the heat cq is determined

by the planimetry method of the field (Fig. 1).

International Letters of Chemistry, Physics and Astronomy Vol. 6 19

Fig. 1. Heat absorbed qc in the system: c – specific heat, t – temperature.

The tables of average specific heat are done usually under foundation of a constant

bottom temperature value, e.g. 01 t C. To calculate the absorbed heat by a body in the

temperature range of 01 t , to 2t , by means of such given the average specific heat, one

should use the formula

102012 tctcq

tt

c (8)

By comparison of the dependences (2), (5), and (8), one obtains the following formula:

12

1020

12

12

2

12

1 tt

tctc

tt

dttc

c

tt

t

tt

t

(9).

As it results from the presented description of the phenomenon of heat absorption by a

system, that description is too simplified, and introduced herewith the magnitudes are

somewhat doubtful. It will be explained on the background of the presented further the

adequate description of the heat absorptions and at the same time transferring the heat by the

system bodies. Thus the quest of using the mechanical-thermal analogy will be raised. It will

allow to clarify the meaning of thermal magnitudes, with the heat as the problematic subject.

3. ADVANTAGE OF MECHANICAL-THERMAL ANALOGY

As it was proved in the former section of the work, the existent definitions of heat have

no adequate character. It appears, the undertaken trials to explain the subject notion have not

allowed to solve that problem/cognitive quest. Furthermore the heat is something which

20 ILCPA Volume 6

penetrates/permeates, passes, traverses, flows, radiates, rises (convection), all in the medium

where the temperature difference occurs. On the grounds of science on the heat, there is a

cognitive gap which should be necessarily filled up with the adequate content of the notion of

this magnitude.

Therefore, the heat is the physical magnitude as it may be measured and provided with

the value. For the time being, that is the way it is characterized. It has been also established,

that the heat is the way of the energy transfer. All it is about the heat energy. Thus there is the

notion of thermal energy, and heat. They are two different notions and none a proper

definition. They are identified, used interchangeably, that leads to the cognitive

pandemonium, if it refers to the determined fragments of the natural reality.

In science, there is the equilibrium of mechanical work and heat, indicated by a known

experiment of Joule (1843). The relationship of heat on the mechanical work is regarded by

the thermal equivalent of mechanical work, and the dependence of the mechanical work on

heat is covered by the mechanical equivalent of heat. Here also, the heat notion is not

explained. The heat is still the non-determined notion.

The heat notion may be explained on the way of reasoning. However, in this case the

cognitive fundamentals, using the philosophic approach and principal methods of the

cognitive work, should be considered. In this case the method of mechanical-thermal analogy,

resulting from the identity of mathematic form of equations describing the mechanical work

and heat, should be applied.

Unfortunately, that analogy has not been noticed as yet, even if the analogy is used,

concerning the problems connected with the heat exchange. For instance, the work [3] turns

the attention on the analogy occurring between the heat exchange and the momentum

exchange, by treating it as the hydromechanic-thermal analogy. The works [7-10], in turn,

explain the use methods of electrical analogy to the studies of flow in the complex elements

of the fluid-flow machines, in this also the ventilators.

At the very beginning of the necessary considerations, concerning the mentioned above

mechanical-thermal analogy, one should state that in fact it is about the thermal work. That is

the source of all other investigations. Coming out of this initial point, one may derive and

explain the notions of the thermal force, thermal power, and thermal potential. The

equivalents are the following notions: mechanical work, mechanical force, mechanical power,

and the mechanical potential.

4. MECHANICAL ESSENCE OF THE WORK, FORCE, POWER, AND

POTENTIAL

The mechanical work is the work of solid, or the body being in the solid state of

aggregation. That general qualitative definition was formed in accordance with the rule of

systematics/satiety [5], which determines the method of creation of all definitions. It is about

the two-order systematics (1 – work, 2 – mechanical work).

That general qualitative definition may be extended respectively. That extended

definition will be the three-order systematics. The third its order forms the closer explanation

which says that the work forms the connection between the reason and effect. Therefore that

three-order systematics has the following configuration: 1 – work, 2 – mechanical work, 3 –

the mechanical work, forming a determined connection between the reason and effect.

Quantitatively, the definition of the mechanical work L says that the physical magnitude

is the product of the mechanical work F and the displacement s of a solid. That force is the

measure of reason, whereas the displacement (path length) is the measure of effect. Therefore


sFL (1)

and in the elemental form

dssFdL (2)

so that

00

00

ss

dssFdLL (3).

Of course, the first form of the formula on work is related to the situation, when the

force is constant, whereas the second one, that elemental, regards the variability of that force

on the length of the way of displacement or the deformed body. That second one, variable

force, may be determined as the unit work, that is

ds

dLsl

(4)

The mechanical power LN is the magnitude generally known. It concerns the work

variability L in time τ, that is the work flux L , then

d

dLLNL

(5)

The presented characteristics are concerned with the mechanical behaviours of the

material body in the space and interstate space-time, that is the matter occurring between the

neighbouring potential fields, where the determined energetic states of the material body take

place.

Now there is the need to introduce the magnitude which will characterize these states.

That is the energy of the material body which is the ability to perform a work, or the readiness

to perform the work over it, at the transition on the neighbouring potential field. The measure

of energy is the potential FV , being the product of the potential field intensity and the distance

10s between the neighbouring potential fields. The field intensity is determined by the initial

force 0F . Therefore

100 sFVF (6)

As it results, the mechanical potential (measure of the mechanical energy) is equivalent

with the mechanical work. However, it does not mean that both these magnitudes are

analogous. There are some determined connections between them, resulting from the relations

22 ILCPA Volume 6

100

0

10

sF

dssF

V

Lk

s

F

L (7)

10

0

100

s

FV

dssF

sF

L

Vk

(8)

which the relations are named the equivalents: equivalent of the mechanical work Lk and the

equivalent of the mechanical potential Vk , respectively.

The notions of the mechanical magnitudes: work, force, power, are generally known,

but the energy (measure of this magnitude, that is the potential) has not been clearly explained

as yet. The introduced herewith the definition has the adequate character and fully reveals the

essence of this notion. That has been used now, reflected by the following exemplary works

[11-17].

5. THERMAL CHARACTER OF THE WORK, FORCE, POWER, AND

POTENTIAL

The mechanical-thermal analogy allows to determine the heat (thermal work) in the

following manner:

dttFdQ (9)

where the differential dQ denotes the elemental thermal work, tF is the thermal force, and

the differential dt is the elemental temperature.

Of course this thermal force is the product of the mass m of a material body

(thermodynamic factor) and its specific heat tc , then

dttcmdQ (10)

so further

1

0

1

0

t

t

t

t

dttcmdttFQ (11)

where 0t is the initial temperature, 1t the final temperature.


In analogy to the previous action, this force is a unit work, that is tcmtF .

Therefore that unit thermal work takes the following form

dt

dQtq (12)

The thermal power QN is the work Q variability, that results from the temperature t

change. This is now the heat flux (of thermal work), so

dt

dQQNQ

(13)

It is time now to explain the notion of thermal energy. First, however, it should be noted

that until now there is no clear univocal definition of that magnitude. Thus let us focus on the

work [18], by J. Miquel Rubi. Therein he states that the thermodynamics is one of such a part

of physics which is understood erroneously. There are many notions in use by different

scientists which do not necessarily reflect the exact meaning and the relevant subtleties. Thus

the author of [18] has mentioned that he touches deep into the theory but expressed his feeling

to be careful in this matter.

How then J. Miquel Rubi understands the energy notion? It is reflected with his

explanation of the phenomenon of thermal equilibrium: „While bringing into contact two

bodies of different temperature, the energy in the form of heat flows from the warmer to

cooler one. It lasts until the equilibrium of temperatures of both bodies is achieved, which is

the thermal equilibrium”.

The presented explanation also does not reveal the true essence of energy. Energy does

not flow and is not a form of heat. They are two quite different magnitudes. These basic

differences are distinctive at the initial stage of their characterization, if the kind of

magnitudes is clarified.

The energy is a mental magnitude, the mind creature, the notion category. Therefore it

is not a physical magnitude, which could be measured and recorded by means of mathematic

tools, to determine its value, price or consumption. One may differentiate the sort of energy

but not its form. All it is about the thermal energy. The heat, or rather thermal work, is the

way of energy transfer, and this transfer has non-continuous, jump, quantum character.

Thus the energy is the function of state. Therefore its presence is detected on the

determined energetic levels only (potential fields). Between these levels/fields, there is no

energy at all. There the intensified temperature variability occurs, proving of that the thermal

work takes place.

Of course, now a determined measure, as the physical magnitude to be undergoing

measurements, should be assigned to the energy (mental magnitude). There is such a measure.

That is a potential, being the product of the potential field intensity and a determined space (or

space-time) resistance, dividing the neighbouring potential fields.

One may now present the adequate definition of the thermal energy. The thermal energy

of a material body (of thermodynamic agent) is its ability to perform the work, or the

readiness to perform the work over it, at the transition onto the neighbouring potential field.

The measure of this energy is the potential FV , being the product of the potential field

intensity and the thermal distance between these fields. The field intensity is the initial

thermal force 0F that is the initial unit thermal work, denoted as 0q . The thermal distance is

24 ILCPA Volume 6

the difference 01 tt of temperatures, which determine the position of the neighbouring

potential fields. That refers to the limits of the definite integral, determining the thermal work.

Therefore

010010 ttmcttFVF (14)

where 0c refers to the initial specific heat of a determined thermodynamic agent.

6. GENERAL CHARACTERISTICS OF HEAT ABSORPTION BY A THERMAL

SYSTEM

The characteristics is the dependence of thermal force F on temperature t, that is

tfF . That force is the product of the heated body mass and its specific heat; thus

tcmtF .

However, at first it is worth presenting the most general form of the source differential

equation. That form is referred to all phenomena and/or processes, having a quantum course

which is changing by a jump. The phenomena are considered in macro-scale. Of course, such

a sort of changes is related to the states, not the courses between the states, because between

them the continuous changes of a determined magnitude take place.

The general differential equation, excerpted from [19], has the following configuration:

dNN

ZdZ

(15)

where: dZ total differential of the dependent variable; dN total differential of the

independent variable; N

Z

partial derivative of the dependent variable, referred to the

independent variable. The signs are the algebraic operators fulfilling a determined role.

The sign has a formal meaning, confirming the physical sense of a determined

dependence. The sign ascribes such a sense to the determined record.

Considered here the phenomenon of heat absorption describes two magnitudes, namely

the thermal force F and temperature t. Thus that force will appear as the independent variable,

and the temperature will take the role of the independent variable. Regarding the fact that the

course of the considered magnitude is exponential and rising degressively, then

dtt

FdF

(16)

The scheme of creation of the adequate description of the dependence tfF is

presented in Fig. 2, revealing all elements of this reasoning process. The curve illustrating that

dependence comes out of the initial point „0” of the coordinates 0tt , 0FF , and further it

has the course rising exponentially and degressively. It completes its non-linear course in the

point „1”, where the body heating phenomenon has its end.


Fig. 2. Indicative course of the dependence of thermal force F on temperature t.

Now the integration of dependence (16) on both sides should be done remembering that

the total differential is the state function. It is needed to determine the states which are the

integration limits of this thermal phenomenon.

The mentioned limits are the potential fields. These fields are situated on two directions;

one of them is the direction of thermal force, whereas the other one is the direction of

temperature. On the first direction, one may find: the bottom force stable potential field

FBSPF , upper force stable potential field FTSPF , force unstable potential field FAPP ,

and the nominal potential field NPF. On the second direction, there are the following fields:

the temperature stable potential field tSPF )( , and temperature unstable potential field

tAPF )( . Between the fields FBSPF and FTSPF there is the potential band (thick dotted

area in Fig. 2). The fields FTSPF , FAPF , tSPF )( and tAPF )( limit/reduce the proper

temperature-force space (sparsely dotted area), where the real exponential and degressively

rising changes of the thermal force occur (solid line). Between the fields FAPF and NPF

26 ILCPA Volume 6

there is an improper space, fulfilling here an auxiliary role (this is also an auxiliary

mathematic tool).

The curve of thermal force, comprised between the points 0-1, is the envelope of right-

angled triangles, moving with its horizontal leg on the nominal potential field NPF; with that

horizontal leg being invariable and equal to the temperature constant T, whereas the vertical

leg varies respectively, decreasing within the displacement of the triangle in the temperature

direction. That nominal field is situated symmetrically against the level of the heating body

phenomenon completion, in the distance from it equal to the length of the real space in the

direction of the thermal force.

Thus over the real/proper space there is the above mentioned the improper space

situated, where the curve (dashed line) approaches the asymptote, being that nominal potential

field. That creature is surely an auxiliary design, needed to the description of the real curve,

reflecting the adequate dependence of the thermal force on temperature.

Now one may proceed with the integration of the equation (16). At integrating of that

equation one should mark the integrals limits of the total differentials. That means

Tt

t

FFF

FF

dtt

FdF

011

0

(17)

and further

Tdt

dFFFF 012 (18)

or

dt

TFFF

dF 1

2 01

(19).

One may notice, the partial derivative has been substituted by the quotient of total

derivatives. It could be done that way because the total differentials have been clearly

determined by introducing the limits of the integrals.

Furthermore, by integrating both sides of the equation (19), one obtains the following

result

*

01

12ln Ct

TFFF (20)

that is

T

t

T

t

CC

T

t

CeeeeFFF

*

*

012 (21)

After regarding that for 00 ttt , that is 0tt , the magnitude 0F , one obtains


T

t

eFFC0

012 (22)

and after substituting (22) to (21), and further, after relating to 0FFF

T

tt

eFFFF0

12 010 (23).

One may now determine the second coordinate of the point „1”, i.e. 0110 ttt .

That is obtained by introducing 1FF and 1tt to the equation (23). Therefore

2ln10 Tt (24)

7. ANALYSIS OF ADEQUATE DEPENDENCE OF THERMAL FORCE ON

TEMPERATURE

The dependence (23) presents a final, analytical form of dependence of the thermal

force F on temperature t. It is also the outcome basic functional thermal characteristics of the

heated thermodynamic agent. Based on this, now further derivative functional characteristics

may be created. Of course this outcome characteristics contain the determined parametric

characteristics, i.e. 0F , 1F , 0t , and T.

The first derivative functional characteristics is the dependence of the increment rate

(tempo, intensity) of thermal force on temperature. Thus

T

tt

FT

tt

F ekeT

FF

dt

dFk

00

0012

(25)

where 0

Fk is the initial rate of the force increment. For 0tt that rate 0

FF kk , and for 1tt ,

i.e. 2ln0110 Tttt , the rate 05.0 FF kk .

The course of this magnitude is illustrated in Fig. 3.

28 ILCPA Volume 6

Fig. 3. Dependence of the increment rate of thermal force on temperature.

The graphic image (Fig. 4 a) of the initial dependence, that is (23), may be the basis to

explain the essence of further derivative magnitudes, that is the thermal work (Fig. 4 b) and

initial (Fig. 4 c) and final thermal potential (Fig. 4d). These potentials are the measures of

thermal energies: initial, and final.

Fig. 4. Illustration of the dependence of force (a), work (b), initial (c), and final (d) thermal potential

on temperature.


The thermal work FL corresponds with the field under the curve tfLF , described

by the formula of the following form:

dteFFFdttFL

t

t

T

ttt

t

F

1

0

01

0

12 010

T

tt

eFFTttF01

122 01011 (26).

The mentioned thermal potentials (initial 0

FV and final 1

FV ) are described by the

following formulae:

010

0 ttFVF (27)

011

1 ttFVF (28).

By regarding further that the thermal force F is the product of the heated body mass and

its specific heat, the formulae (23), (25), (26), (27), (28) assume the following configurations:

T

tt

ecccmF0

12 010 (29)

T

tt

FT

tt

F ekeT

ccmk

00

0012

(30)

T

tt

F eccmTttmcL01

122 01011 (31)

010

0 ttmcVF (32)

011

1 ttmcVF (33).

By recording in the formula (29) the force F as a product cm , one obtains as follows:

30 ILCPA Volume 6

T

tt

ecccmcm0

12 010 (34)

that means

T

tt

ecccc0

12 010 (35)

that is the dependence of the specific heat on temperature.

8. CONCLUSION

The problem of heat absorption through a thermal system has been considered. Apart

from the existent notions ordering, quite new notions have been introduced. It appears, the

existent notion of heat was identified as the heat potential being the measure of heat energy.

The use of mechanical-thermal analogy made it possible to explain several terminology

quests. Thus the set of characteristics of the heat absorption phenomenon consists of: the

thermal force, heat work, thermal potential, rate/intensity of increment of thermal force, and

the specific heat.

The outcome of source, that is of the starting differential equation, being differential

dependence of the thermal force on temperature, made it possible to achieve the analytic,

functional characteristics of the title phenomenon. That characteristics, being the causal model

of the described natural system, has a deep physical reasoning.

It is worth stating that such an approach, coming out of the source differential equation,

has been used by the authors earlier [20,21]. In those cases quite a complex dependence of the

tool life under cut on the main velocity in the machining system, was used [20] and the

kinetics of tool edge fixed flexibly was solved [21].

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[2] M. Jeżewski, Physics, PWN, Warszawa 1966 (in Polish).

[3] B. Staniszewski, Heat exchange. Theoretical foundations, PWN, Warszawa 1980

(in Polish).

[4] S. Ochęduszko, Applied thermodynamics, WNT, Warszawa 1967 (in Polish).

[5] M. Mazur, Technical terminology, WNT, Warszawa 1961 (in Polish).

[6] S. Ochęduszko, J. Szargut, H. Górniak, A. Guzik, S. Wilk, Exercises of technical

thermodynamics PWN, Warszawa 1968 (in Polish).

[7] T. Chmielniak, A. Szafraniec, Przegląd Mechaniczny 2 (1977) 41-45 (in Polish).


[8] M. Hackeschmidt, Strömungstechnik-Ähnlichkeit. Analogie-Modell. Lipsk 1972.

[9] M. Hackeschmidt M., Die Elektroanalogie ein Mittel zur Lösung komplizierter

Feldprobleme. Habilitationsschrift, Technische Universität, Dresden 1964/65.

[10] M. Hackeschmidt, Zur Ermittlung der reibungsfreien Strömung durch rotierende

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[11] Tadeusz Hryniewicz, Zdzisław Pluta, Understanding the Teaching Process or the Lost

Essence of Energy. Proc. of the 8th

Word Conference on Continuing Engineering

Education, Theme A.: Knowledge, Skills and Competency, May 12-16, 2001, Toronto,

Ontario, Canada, pp. 25-31.

[12] Zdzisław Pluta, LAB 4 (2005) 43-46 (in Polish).

[13] Zdzisław Pluta, Energetyka 7 (2005) 496-501 (in Polish).

[14] Zdzisław Pluta, Return to the sources, Forum Akademickie 3 (1999) 58-60 (in Polish)

[15] Zdzisław Pluta, Energetic aspects of cutting using single abrasive grain fixed flexibly.

Proc. 23rd Sci. School on Abrasive Treatment. Rzeszów-Myczkowice, September

2000, pp. 303-309 (in Polish).

[16] Zdzisław Pluta, Potential and the work of a single abrasive grain, Proc. 25th

Sci. School on Abrasive Treatment. Wrocław-Duszniki Zdrój, 11-14 September 2002,

pp. 389-394 (in Polish).

[17] Zdzisław Pluta, Determination of energetic potential of geometrically compact abrasive

tools, Proc. 25th Sci. School on Abrasive Treatment. Wrocław-Duszniki Zdrój, 11-14

September 2002, pp. 395-400 (in Polish).

[18] J. Miquel Rubi, Świat Nauki (208)12 (2008) 44-49 (in Polish).

[19] Zdzisław Pluta, Fundamentals of surface smoothing by elastic abrasive wheels,

Wydawnictwo Politechniki Koszalińskiej, Koszalin 2007 (in Polish).

[20] Zdzisław Pluta, Tadeusz Hryniewicz, Int. J. Adv. Manuf. Technol. 51 (2010) 217-223,

DOI:10.1007/s 00170-010-2595-5.

[21] Zdzisław Pluta, Tadeusz Hryniewicz, Int. J. Adv. Manuf. Technol. 62(5) (2012)

529-542, DOI:10.1007/s 00170-011-3813-5.

32 ILCPA Volume 6

heat absorption by a thermal system

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